Rundown mechanic

[JJ]

I posed both scenarios to some other umpires, and the concensus seems to be the PU has the rundown, but the BU can "fill in" IF he communicates this clearly to the PU. That's much like the rundown betweeen first and second when R1 gets picked off - BU has the whole rundown until the PU gets to first base and lets him know he's there to help out.
As to the second question posed, I'd have to say the BU has the long distance call at third UNTIL the runner gets himself in a rundown. Then the PU has the rundown. If they want a better look, hire the third guy...
All just my opinion, FWIW.